I have also wondered about question (i) in the past, and fortunately the answer is yes. Here is a reference:

Verdier, Stratification de Whitney et theoreme de Bertini-Sard, Invent 1976, Cor 5.1

The result is probably also contained in Thom, Bull AMS 75 (1969), but it may be harder to extract (at least it was for me).

Wouldn't (iii) should follow from (i) + Stein factorization, or is there something that I'm missing?

I have also wondered about question (i) in the past, and fortunately the answer is yes. Here is a reference:

Verdier, Stratification de Whitney et theoreme de Bertini-Sard, Invent 1976, Cor 5.1

The result is probably also contained in Thom, Bull AMS 75 (1969), but it may be harder to extract (at least it was for me).

Wouldn't (iii) follow from (i) + Stein factorization, or is there something that I'm missing? [In rereading your question, I realized you posed this only in the event that (i) failed.]